Linearized Domain Decomposition Approaches for Boundary Value Problems

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Linearized Domain Decomposition Approaches for Boundary Value Problems

by

⃝c Faysol Ahmed B. Sc.(Hons)

A thesis submitted to the School of Graduate Studies

in partial fulfillment of the requirements for the degree of

Master of Science.

Scientific Computing

Memorial University of Newfoundland

September 24, 2015

St. John’s Newfoundland

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Abstract

The purpose of this study is to analyze linearized domain decomposition approaches for different nonlinear boundary value problems (BVPs). Nonlinear BVPs frequently form a large system of equations when they are discretized and require parallel computers to solve this system. Domain decomposition approaches are useful to utilize the advantages of parallel computers in order to solve the differential equations. Cher- pion’s single domain linearized iterative technique is quite useful to solve the nonlinear BVPs that have the form u^′′ =f(ξ, u, u^′). However with this iterative scheme we are not able to solve the BVP using parallel computers. Therefore we extend this iterative scheme to the domain decomposition context so that we can solve the nonlinear BVP on parallel computers. Theoretical and numerical results are given.

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Acknowledgements

I am grateful to my supervisor, Dr. Ronald Haynes for his continuous support, proper guidance and encouragement. I could not have made this journey without his support. Furthermore, I am grateful to the School of Graduate Studies for supporting me financially throughout my degree.

Throughout my academic life I have been in touch with many great teachers;

I express my special gratitude to them. I am thankful to my parents and my grand- mother for always being there for me. Finally, I thank all my friends for their love and inspiration.

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List of Figures

2.1 Staggered grid discretized.. . . .8

2.2 Decomposition into two subdomains.. . . .11

2.3 Domain decomposed into several subdomains.. . . .11

2.4 Order of the local truncation error using midpoint formula to approximate the mesh density function.. . . .13

2.5 Order of the local truncation error using trapezoidal formula to approximate the mesh density function.. . . .14

2.6 Order of discretization both the midpoint formula and the trapezoidal formula.. . . .14

2.7 Rate of convergence of the Newton’s method.. . . .15

3.1 Contradictory shape of function w.. . . .28

3.2 Sample shape of function z1.. . . .30

3.3 Domain partition for P(ξ). . . .34

3.4 u^′ not bounded above by R.. . . .42

3.5 u^′ not bounded below by −R.. . . .43

3.6 u^′ not bounded above by R.. . . .45

3.7 u^′ not bounded below by −R.. . . .47

4.1 Domain decomposed into two subdomains.. . . .71

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4.2 Iteration starting from the subsolution.. . . .79 4.3 Iterations starting from the supersolution.. . . .88

5.1 DD error vs number of iterations for different overlap for the 1st subdomain for BVP (5.1).. . . .114 5.2 DD error vs number of iterations for different overlap for the 2nd sub-

domain for BVP (5.1).. . . .115 5.3 DD error vs iteration for different numbers of subdomains for BVP (5.1).115 5.4 Plot of the exact solution of BVP (5.2).. . . .117 5.5 Linearized iterations starting from the subsolution for BVP (5.2).. .118 5.6 Linearized iterations starting from the supersolution for BVP (5.2)..118 5.7 Monotonicity of the iterates for BVP (5.2).. . . .119 5.8 Plot of the analytic solution of BVP (5.5).. . . .120 5.9 Linearized Iterations starting from the subsolution for BVP (5.5).. .121 5.10 Linearized iterations starting from the supersolution for BVP (5.5)..121 5.11 Monotonicity of iterations for the solution of BVP (5.5).. . . .122 5.12 Plot of the analytic solution of BVP (5.5).. . . .123 5.13 Linearized Iterations starting from the subsolution for BVP (5.8).. .124 5.14 Linearized iterations starting from the supersolution for BVP (5.8)..124 5.15 Monotonicity of iterations for the solution of BVP (5.8).. . . .125 5.16 Linearized iterations starting from the subsolution for BVP (5.11).. .126 5.17 Linearized iterations starting from the supersolution for BVP (5.11)..126 5.18 Monotonicity of iterations for BVP (5.11).. . . .127 5.19 Plot of the analytic solution of BVP (5.12).. . . .128 5.20 Linearized iterations starting from the subsolution for BVP (5.12).. .129 5.21 Linearized iterations starting from the supersolution for BVP (5.12)..129

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5.22 Monotonicity of iterations for BVP (5.12).. . . .130 5.23 Plot of the analytic solution for BVP (5.15).. . . .131 5.24 Linearized iterations starting from the subsolution for BVP (5.15).. .132 5.25 Linearized iterations starting from the supersolution for BVP (5.15)..132 5.26 Monotonicity of iterations for BVP (5.15).. . . .133 5.27 Numerically calculated exact solutions of BVP (5.18).. . . .133 5.28 Linearized iterations starting from the subsolution for BVP (5.18).. .134 5.29 Linearized iterations starting from the supersolution for BVP (5.18)..135 5.30 Monotonicity of the iterates for BVP (5.18).. . . .135 5.31 Linearized DD iterations starting from the subsolution for BVP (5.21).136 5.32 Linearized DD iterations starting from the supersolution for BVP (5.21).137 5.33 Monotonicity of iterates starting from subsolution for BVP (5.21).. .137 5.34 Monotonicity of iterates starting from supersolution for BVP (5.21)..138 5.35 Effect of the overlap for the linearized DD solution of BVP (5.21).. .138 5.36 Effect of the number of subdomains on the linearized DD solution of

BVP (5.21).. . . .139 5.37 Relation between iterates (4.2) and (4.3) starting from subsolution of

BVP (5.21) for n= 9.. . . .139 5.38 Relation between iterates (4.2) and (4.3) starting from supersolution

of BVP (5.21) for n= 9.. . . .140 5.39 Linearized DD iterations starting from the subsolution for BVP (5.22).141 5.40 Linearized DD iterations starting from the supersolution for BVP (5.22)141 5.41 Monotonicity of the iterates starting from subsolution for BVP (5.22).142 5.42 Monotonicity of the iterates starting from supersolution for BVP (5.22).142 5.43 Effect of the overlap on the linearized DD solution of BVP (5.22).. .143

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5.44 Effect of the number of subdomains on the linearized DD solution of BVP (5.22).. . . .143 5.45 Relation between iterates (4.2) and (4.3) starting from subsolution of

of BVP (5.22) for n= 9.. . . .145 5.47 Linearized DD iterations starting from the subsolution for BVP (5.23).146 5.48 Linearized DD iterations starting from the supersolution for BVP (5.23).146 5.49 Monotonicity of the iterates starting from subsolution for BVP (5.23).147 5.50 Monotonicity of the iterates starting from supersolution for BVP (5.23).147 5.51 Effect of the overlap on the linearized DD solution of BVP (5.23).. .148 5.52 Effect of number of subdomains on the linearized DD solution of BVP

(5.23).. . . .148 5.53 Relation between iterates (4.2) and (4.3) starting from subsolution of

of BVP (5.23) for n= 9.. . . .150 5.55 Linearized DD iterations starting from the subsolution for BVP (5.24).151 5.56 Linearized DD iterations starting from the supersolution for BVP (5.24).151 5.57 Monotonicity of iterates starting from subsolution for BVP (5.24).. .152 5.58 Monotonicity of iterates starting from supersolution for BVP (5.24)..152 5.59 Effect of the overlap on the linearized DD solution of BVP (5.24).. .153 5.60 Effect of the number of subdomains on the linearized DD solution of

BVP (5.24).. . . .153 5.61 Plot showing inequality (5.25) for BVP (5.24) for n = 9.. . . .154 5.62 Plot showing inequality (5.26) for BVP (5.24) for n = 9.. . . .155

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5.63 Linearized DD iterations starting from the subsolution for BVP (5.27).156 5.64 Linearized DD iterations starting from the supersolution for BVP (5.27).156 5.65 Monotonicity of iterates starting from subsolution for BVP (5.27).. .157 5.66 Monotonicity of iterates starting from supersolution for BVP (5.27)..157 5.67 Effect of the overlap on the linearized DD solution of BVP (5.27).. .158 5.68 Effect of the number of subdomains on the linearized DD solution of

BVP (5.27).. . . .158 5.69 Plot showing inequality (5.25) for BVP (5.27) for n = 9.. . . .159 5.70 Plot showing inequality (5.26) for BVP (5.27) for n = 9.. . . .159 5.71 Linearized DD iterations starting from the subsolution for BVP (5.15).160 5.72 Linearized DD iterations starting from the supersolution for BVP (5.15).161 5.73 Monotonicity of iterates starting from subsolution for BVP (5.15).. .161 5.74 Monotonicity of iterates starting from supersolution for BVP (5.15)..162 5.75 Plot showing inequality (5.25) for BVP (5.15) for n = 9.. . . .162 5.76 Plot showing inequality (5.25) for BVP (5.15) for n = 9.. . . .163 5.77 Linearized DD iterations starting from the subsolution for BVP (5.29).164 5.78 Linearized DD iterations starting from the supersolution for BVP (5.24).164 5.79 Monotonicity of iterates starting from subsolution for BVP (5.29).. .165 5.80 Monotonicity of iterates starting from supersolution for BVP (5.29)..165 5.81 Effect of the overlap on the linearized DD solution of BVP (5.29).. .166 5.82 Effect of the number of subdomains on the linearized DD solution of

BVP (5.24).. . . .166 5.83 Plot showing inequality (5.25) for BVP (5.24) for n = 9.. . . .167 5.84 Plot showing inequality (5.31) for BVP (5.29) for n = 9.. . . .168 5.85 Iterations starting from the subsolution for the mesh BVP (5.32).. .169 5.86 Iterations starting from the supersolution for the mesh BVP (5.32)..170

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5.87 Monotonicity of the iterates for BVP (1.6).. . . .170 5.88 Linearized DD iterates starting from subsolution for the mesh BVP

(5.32).. . . .171 5.89 Linearized DD iterates starting from supersolution for the mesh BVP

(5.32).. . . .171 5.90 Monotonicity of iterates starting from subsolution for the mesh BVP

(5.32).. . . .172 5.91 Monotonicity of iterates starting from supersolution for the mesh BVP

(5.32).. . . .172 5.92 Plot showing inequality (5.25) for the mesh BVP (5.32).. . . .173 5.93 Plot showing inequality (5.26) for the mesh BVP (5.32).. . . .174

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Chapter 1 Introduction

We are motivated by solving steady state physical problems of the form

L{u}= 0 u(0) =a, u(1) =b, (1.1) where L is a spatial differential operator. If this physical problem has a “difficult”

solution, then solving this using a uniform mesh will not give us an accurate result.

To solve this problem in a non-uniform physical coordinate x, we transform this physical coordinate to a new computational uniform coordinate ξ, x = x(ξ), where x(0) = 0, x(1) = 1 andξ∈[0,1]. We want to solve the differential equation efficiently for the variableξ using as few points as possible. Often we wish to use a uniform grid

ξi =ih, i= 0,1, ..., N.

A standard way to do this is to apply the equidistribution principle. Suppose M(x, u) is the measure of difficulty or error in the solution of the physical problem. We choose xi, i= 0,1,2, ..., N, so that

∫ xi

xi−1

M(˜x, u)d˜x≡ 1 N

∫ 1 0

M(˜x, u)d˜x. (1.2)

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The above equation indicates that the error in each subdomain is equal to the average error in the whole domain. Here ∫1

0 M(˜x, u)d˜x is the total error and _N¹ ∫1

0 M(˜x, u)d˜x is the average error. A continuous mesh transformation between the computational coordinate ξ and the physical coordinate x can be achieved by adding (1.2) up toi,

∫ x_i 0

M(˜x, u)d˜x= i N

∫ 1 0

M(˜x, u)d˜x, and by introducing the continuous variable ξ we have

∫ x(ξ) 0

M(˜x, u)d˜x=ξ

∫ 1 0

M(˜x, u)d˜x. (1.3)

Differentiating (1.3) twice with respect to ξ, we obtain d

dξ [

M(x(ξ), u) d dξx(ξ)

]

= 0. (1.4)

The above differential equation is nonlinear with Dirichlet boundary conditionsx(0) = 0 and x(1) = 1. After forming the equation (1.4) we assemble the coupled system of differential equations

L{u}= 0 u(0) =a, u(1) = b, (1.5) d

dξ {

}

= 0 x(0) = 0, x(1) = 1. (1.6) We can solve this coupled system simultaneously by considering mesh equation (1.6) and physical PDE (1.5) as one large system. This approach is relatively simple and we can solve the coupled system directly by discretizing the whole system and then forming it into one big system of equations. However this approach is not efficient as the simultaneous solution has a highly nonlinear coupling between the physical solution and the mesh. Another disadvantage is that this large system loses the properties that the physical partial differential equation (PDE) and mesh equation may individually have. We can also solve this coupled system through an alternating

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solution procedure, referred to as the MP procedure [12]. In this iterative approach initially we chose a mesh xⁿ⁻¹ to solve the physical equation. With this mesh we solve for an approximate physical solution then with this physical solution we solve the mesh equation forxⁿ. We will repeat this iteratively until we get the desired level of accuracy in the solution of physical equation. The advantages of this approach are its flexibility and efficiency. In this procedure we can utilize the features of the mesh equation in order to solve the boundary value problem (BVP) (1.6) more efficiently.

This mesh equation can be solved using one of the following algorithms: De Boor’s algorithm, an algorithm based on MMPDE5xi or one based on direct optimiza- tion of some error bound [12]. However these methods are not suitable for parallel computing. To take advantages of parallel computing, one suitable method of solving this mesh BVP is to use domain decomposition. Domain decomposition (DD) is based on a divide and conquer philosophy. The DD is a technique, which divides a big problem into several subproblems on smaller overlapping or non-overlapping subdomains which form a partition of the original domain. In our analysis we will only consider overlapping subdomains. For overlapping subdomains we can either solve this mesh BVP by the nonlinear domain decomposition method or a linearized domain decomposition method. The main advantage of the nonlinear domain decomposition method is that it provides a fast convergent solution. However, at the same time, the drawback of this method is that in each iteration the solution of many nonlinear systems of equations are required. The linearized domain decomposition method does not have this problem. Motivated by this nonlinear mesh BVP we de- velop a linearized domain decomposition method which can solve the BVPs of the form u^′′ = f(ξ, u, u^′), where f(ξ, u, u^′) depends on u^′ nonlinearly. The methods developed will be a first step towards a viable linearized domain decomposition method

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for the mesh BVP.

Linearized methods for nonlinear BVPs have a long history. A linear monotone iterative approach was given by Lui [14] to solve BVPs of the form u^′′ = f(ξ, u) in a convenient way. Cherpion [6] stated and proved another linearized iterative approach on a single domain to solve the BVPs which have the form u^′′ = f(ξ, u, u^′).

We have extended the iteration scheme from [6] to a domain decomposition approach.

Monotonic iterative approaches starting from subsolutions and supersolutions for the BVP of the form (1.7)

−u^′′+f(ξ, u) = 0, u(a) = u(b) = 0. (1.7) were first introduced by Picard in 1893 [16]. After this work many other monotone iterative methods were developed to solve this type of BVP. In 1931, using subsolutions and supersolutions, G. Scorza Dragoni [7] proved that solutions exist for the BVPs of the form

−u^′′+f(ξ, u, u^′) = 0, u(a) =u(b) = 0. (1.8) He assumedf was continuous and bounded to prove the existence of solution of (1.8).

Dragoni modified (1.8) by replacing f by ¯f, where ¯f = f between the subsolution and supersolution. With this choice, he easily controlled the nonlinear dependency of the derivative. He proved that the solution of the modified BVP exists and then using the maximum principle he proved that this solution of the modified BVP is the solution of (1.8). In 1964, Gendzojan [9] developed the monotone iterative methods for the BVP of the form (1.8). This is the very first monotone iterative approach that does not impose any constraint on the nonlinear dependency on the derivative. For

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a given subsolution α0 and supersolution β0 he considered the iteration schemes for n= 1,2,3, ...,

−α^′′_n+K(ξ)α^′_n+l(ξ)αn =−f(ξ, αn−1, α^′_n−1) +K(ξ)α^′_n−1 +l(ξ)αn−1, (1.9) α_n = 0 on∂Ω,

−β_n^′′+K(ξ)β_n^′ +l(ξ)βn =−f(ξ, βn−1, β_n−1^′ ) +K(ξ)β_n−1^′ +l(ξ)βn−1, (1.10) βn = 0 on∂Ω.

Here the functions K(ξ) and l(ξ) depend on the assumptions on the function f. For this reason iteration schemes (1.9) and (1.10) are not feasible from the computational point of view. In 1974, J. Chandra and P.W. Davis [4] developed an iterative method to solve a problem that depends linearly in the derivative. Following that in 1977, S.R.

Bernfeld and J. Chandra [1] generalized this method for the problem that depends on the derivative nonlinearly. They considered iterations of the form

−α^′′_n+M αn=−f(ξ, αn−1, α^′_n) +M αn−1, α_n= 0 on ∂Ω,

−β_n^′′+M βn =−f(ξ, βn−1, β_n^′) +M βn, βn= 0 on ∂Ω,

provided that α0 and β0 are subsolution and supersolution of BVP (1.8) respectively and M is a constant based on the assumption of f. As α^′_n and β_n^′ appear explicitly on the right side of the scheme, explicit solutions are not possible with these iterates.

Thereafter, in 2001 M. Cherpion [6] proposed another iteration scheme which is more

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simple and computationally feasible. The iteration scheme is given by α^′′_n−√³

lK(ξ)α^′_n−lα_n =f(ξ, α_n−1, α^′_n₋₁)−√³

lK(ξ)α^′_n₋₁−lα_n−1, (1.11) αn= 0 on ∂Ω,

β_n^′′−√³

lK(ξ)β_n^′ −lβ_n =f(ξ, β_n−1, β_n^′₋₁)−√³

lK(ξ)β_n^′₋₁−lβ_n−1, (1.12) βn= 0 on ∂Ω,

whereα0 and β0 are subsolution and supersolution of BVP (1.8) respectively, K(ξ) is an antisymmetric function on Ω and l >0 is a constant depends on the assumptions of f.

As an example of domain decomposition methods for nonlinear problem the second chapter provides a discussion of the nonlinear domain decomposition method used to solve the moving mesh problem which was published in [8]. The third chapter introduces the existing work [6] of Cherpion’s linearized iterative approach on a single domain to solve the BVP of the formu^′′ =f(ξ, u, u^′). We also present a single domain analysis of the iterations suggested by S.H Lui’s DD approach [14] to solve problems of the form u^′ =f(ξ, u). The fourth chapter contains S.H Lui’s DD approach from [14]

and a new DD version of Cherpion’s iteration is demonstrated and analyzed. Finally the fifth chapter provides the numerical results to support the theory.

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Chapter 2 Nonlinear Single Domain and

Domain Decomposition methods

To give a flavor of the numerical and parallel approaches to solve boundary value problems we will use BVP (1.6) as a model problem. In this chapter we will discuss nonlinear single domain method and domain decomposition method to solve the mesh BVP (1.6). We will present the finite difference formulation for the nonlinear single domain method.

2.1 Single Domain Approach to solve the BVP

Our focus will be on nonlinear two point boundary value problems. For example the BVP for the mesh maybe written as,

d dξ

{

}

= 0 (2.1)

x(0) = 0, x(1) = 1.

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Here we demonstrate a typical approach to solve nonlinear BVPs numerically. A standard way to solve (2.1) on ξ∈[0,1] is to use the finite difference approximation.

In order to solve this BVP we will discretize using a staggered grid. Let w(ξ, x) = M(ξ, x) d

dξx then equation (2.1) becomes dw

dξ = 0.

Letxj approximatex(ξj), whereξj =jh, j = 0,1, ..., N. Now we approximatew(ξ, x)

0

ξ_j₋¹

2 ξ_j+¹

2

ξj−1 ξj ξj+1 1 ξ

Figure 2.1: Staggered grid discretized.

by taking a short difference at ξ_j+¹

2 and ξ_j−¹

2. Letw_j+¹

2 and w_j−¹

2 be approximations of w atξ_j+¹

2 and ξ_j₋¹

2 respectively, we have w_j+¹

2 ≈M(x_j−¹

2)x_j+1−x_j

h ,

and

w_j−¹

2 ≈M(x_j+¹

2)xj−xj−1

h .

Hence we get the approximation to (2.1) at ξj as 1

h [w_j+¹

2 −w_j₋¹

2

]= 0. (2.2)

Now we may approximate M at ξ_j+¹

2 and ξ_j₋¹

2 using the midpoint formula M(x_j+¹

2)≈M

(xj+1+xj

2 )

, and

M(x_j−¹

2)≈M

(x_j +x_j−1 2

) .

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Using these values in equation (2.2), we get the system of discrete algebraic equations M

(xj+1+xj

2 )

(xj+1−xj)−M

(xj +xj−1

2 )

(xj −xj−1) = 0, (2.3) j = 1,2, ..., N.

x0 = 0, xN = 1.

Instead of approximating M at ξ_j+¹

2 and ξ_j₋¹

2 using midpoint formula we may use the trapezoidal rule

M(x_j+¹

2)≈ M(xj+1) +M(xj) 2

and

M(x_j₋¹

2)≈ M(xj) +M(xj−1)

2 .

Then equation (2.2) becomes (M(x_j+1) +M(x_j)

2

)

(xj+1−xj)−

(M(x_j) +M(x_j−1) 2

)

(xj −x_j−1) = 0, (2.4) j = 1,2, ..., N,

x0 = 0, xN = 1.

In both cases we have a nonlinear system of equations. We may write these equations as F(x) = 0 where F :R^N−1 → R^N⁻¹ and the j^th component ofF is given by

Fj(xj−1, xj, xj+1) = 0, j = 1,2, ..., N.

We can solve this nonlinear system using Newton’s method. Given an initial guess x⁰, the Newton update is given by

xⁿ⁺¹ =xⁿ− (∂F

∂x(xⁿ) )−1

F(xⁿ), n= 0,1,2, ... .

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The Jacobian is given by

∂F

∂x =

⎡

⎢

⎣

∂F1

∂x1

∂F1

∂x2 . . . _∂x^∂F¹

N

∂F2

∂x1

∂F2

∂x2 . . . _∂x^∂F_N²

. . .

∂F_N

∂x1

∂F_N

∂x2 . . . ^∂F_∂x^N

N

⎤

⎥

⎦ .

If the initial guess x⁰ is sufficiently accurate and the inverse of the Jacobian ma- trix exists, Newton method will converge quadratically [3], that is error will reduce quadratically which is illustrated in Figure2.7. The order of the local truncation error for (2.4) isO(△x²) as we have used a second order discretization. This is shown in Figure2.4and2.5.

2.2 Nonlinear Domain Decomposition methods

Nonlinear domain decomposition method is a well known technique to solve differential equations on large scale parallel computers. This method is based on divide- and-conquer philosophy. Basically it splits the problem into several sub problems and solves these sub problems independently by adapting boundary conditions on the interfaces. Here only overlapping subdomains are considered for the nonlinear DD approach. On each subdomain a nonlinear BVP with Dirichlet boundary conditions is solved. In 2012, Haynes and Gander [8] solved the mesh BVP by a parallel nonlinear Schwarz method. Here we will discuss their approaches.

Suppose the domain Ω = (0,1) is divided into two subdomains Ω1 = (0, β) and

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Ω2 = (α,1) as in Figure2.2, where α is strictly less than β.

0

Ω₁

Ω2

α βξ 1

Figure 2.2: Decomposition into two subdomains.

For a given x⁰₁(α), x⁰₂(β), we iterate for n = 1,2,3, ...

(M(xⁿ₁)xⁿ_1,ξ)

ξ = 0, ξ∈Ω1, (

M(xⁿ₂)xⁿ_2,ξ)

ξ = 0, ξ ∈Ω2, (2.5) xⁿ₁(0) = 0, xⁿ₁(β) =xⁿ⁻¹₂ (β), xⁿ₂(α) = xⁿ⁻¹₁ (α), xⁿ₂(1) = 1.

If the monitor function M(x) is differentiable and bounded between zero and infinity then the BVP (2.1) has a unique solution [8]. Under these assumptions on the monitor function, iteration (2.5) converges for any initial guesses x⁰₁(α),x⁰₂(β). These statements were proved in [8].

To take the advantages of parallel computer, we assume that the domain is divided into m > 2 subdomains that is Ω_i = (α_i, β_i), i = 1,2, ..., m as shown in Figure2.3.

Ω₁ Ω_m

Ω₂ Ω_i Ωm−1

β1

β2

β_i−1

βi

β_m−2

β_m−1

βm = 1 ξ α1 = 0

α2

α3

αi

αi+1

αm−1

αm

Figure 2.3: Domain decomposed into several subdomains.

Then for given x⁰_i(αi),x⁰_i(βi) the iterations are defined for n= 1,2,3, ... as (M(xⁿ_i)xⁿ_i,ξ)

ξ = 0, xⁿ_i(αi) =xⁿ_i₋⁻₁¹(αi), xⁿ_i(βi) = xⁿ_i+1⁻¹(βi), ξ∈Ωi, (2.6)

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wherei= 1,2,3, ..., m. To ensure the nonadjacent subdomains are not overlapped we assume that βi < αi+2 fori= 1,2,3, ..., m−2. Once the problems are solved in these subdomains we can obtain the whole solution by merging these subdomain solutions.

Suppose the monitor function is differentiable and bounded between zero and infinity, then the classical Schwarz iteration (2.6) converges globally on an arbitrary number of subdomains [8]. Numerical illustrations of these statements are provided in Section 5.1.2. We will also observe if we increase the overlap, the DD method will give us faster convergence. Moreover we will see that as the number of subdomains increases the DD method converges more slowly. However there exists many physical problems, where overlapped domain decomposition is not possible. In this situation the classical Schwarz method will not provide convergent solution. We can also get a convergent solution for these types of problems with an optimal Schwarz method or optimized Schwarz method. Optimal Schwarz method can provide a convergent solution in two iterations for two subdomains [8]. However optimal conditions for optimal Schwarz method are not cost effective to use. Optimized Schwarz method can used efficiently providing much faster convergence than classical Schwarz.

2.3 Brief Numerical Remarks

2.3.1 Order of discretization

We consider the mesh density function M(x) = 1 +x² to analyze the numerical results of nonlinear single domain approach. Figure2.4illustrates the order of the local truncation error in the discretization of two point boundary value problem (2.1), where the mesh density function is discretized using midpoint formula.

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−4.5 −4 −3.5 −3 −2.5 −2

−13

−12.5

−12

−11.5

−11

−10.5

−10

−9.5

−9

−8.5

−8

log(h)

log(Error)

Calculated plot Known slopte 2 line

Figure 2.4: Order of the local truncation error using midpoint formula to approximate the mesh density function.

In Figure2.4the blue line shows that the order of the discretization is two, as this line is parallel to a line with known slope of two. Instead of midpoint formula if we use trapezoidal rule to discretize the mesh density function we will also get second order accurate result. Also for each value of h the error is smaller for midpoint than for trapezoidal. From Figure2.5we clearly observe that order of the discretization is two for the trapezoidal rule. Although discretization of mesh density function using midpoint formula and trapezoidal rule both give second order accurate results, the midpoint method provides better results.

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−4.5 −4 −3.5 −3 −2.5 −2

−12

−11.5

−11

−10.5

−10

−9.5

−9

−8.5

−8

−7.5

log(h)

log(Error)

Calculated plot Known slopte 2 line

Figure 2.5: Order of the local truncation error using trapezoidal formula to approximate the mesh density function.

−4.5 −4 −3.5 −3 −2.5 −2

−13

−12

−11

−10

−9

−8

−7

log(h)

log(Error)

Trapezoidal Mid Point

Known slope 2 line

Figure 2.6: Order of discretization both the midpoint formula and the trapezoidal formula.

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2.3.2 Rate of Convergence of Newton’s method

Theory states that Newton’s method will converge quadratically provided that the initial guess is close enough to the true solution and n is large. That is ϵn+1 ≈ Cϵ²_n where C is a positive constant and ϵn is the error at nth iteration. This implies that if we plot logϵn+1 against logϵn for n = 1,2, ... it will be a slope two line. We certainly observe from Figure2.7that the rate of convergence of Newton’s method is two. When the mesh density function is discretized using the midpoint formula.

−12 −10 −8 −6 −4 −2

−25

−20

−15

−10

−5

log(Error(1:k−1))

log(Error(2:k))

Order of the method Known slope 2 line

Figure 2.7: Rate of convergence of the Newton’s method.

So far we have discussed the nonlinear single domain method and domain decomposition method to solve the BVP. In next chapter we will discuss linearized iterates to solve nonlinear BVPs.

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Chapter 3 Linearized Single Domain Iterations

In the previous chapter we have seen that the nonlinear single domain method required a good initial guess and also it required the solution of systems of nonlinear equation in each iteration. However linearized single domain methods do not require the solution of nonlinear systems of equations in each iteration. Furthermore the initial guess does not have to be close to the true solution. In this chapter we will try to modify Lui’s linearized DD iteration from [14] to a single domain and elaborately explain Cherpion’s linearized single domain iteration from [6] to solve the BVP.

3.1 Linearized single domain method to solve u

^′′

= f (ξ, u)

Consider the PDE

−∆u=f(ξ, u) on Ω, u=h on ∂Ω. (3.1)

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S.H. Lui stated and proved an iterative approach in [14] to solve this kind of BVP for several subdomains. We will review his domain decomposition analysis in Section 4.1. Here we analyze a similar iteration on a single domain.

We begin with same necessary definitions. The subsolution and supersolution of (3.1) are defined in Definition3.1.1and Definition3.1.2.

Definition 3.1.1 A function u∈ C²(Ω) is a subsolution of the PDE (3.1) if

−∆u−f(ξ,u)≤0 on Ω and u≤h on ∂Ω. (3.2)

Definition 3.1.2 A function u¯∈ C²(Ω) is a supersolution of the PDE (3.1) if

−∆¯u−f(ξ,u)¯ ≥0 on Ω and u¯≥h on ∂Ω. (3.3) Now we will define a sector of smooth functions which will be needed in the theorem.

Definition 3.1.3 Suppose u is a subsolution and u¯ is a supersolution of (3.1) with u ≤ u¯ on Ω. Let X = C^µ( ¯Ω)∩ C²(Ω) for some 0< µ < 1, where C^µ( ¯Ω) is the space of H¨older continuous functions on Ω. Define the sector of smooth functions between¯ u and u¯ as

A={u∈ X |u≤u≤u¯}. (3.4) Furthermore an assumption for the theorem is stated in Assumption1.

Assumption 1 Assume f is a H¨older continuous function defined on Ω¯ × A. In addition, suppose there exists some non-negative function c ∈ C^µ( ¯Ω) so that for all ξ ∈Ω and v ≤u∈ A

−c(ξ)(u−v)≤f(ξ, u)−f(ξ, v). (3.5)

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Under this assumption on f for a given subsolution u or supersolution ¯u with u⁰ =u oru⁰ = ¯u we consider the iteration scheme: for n= 0,1,2, ...

−∆u⁽ⁿ⁺¹⁾+cu⁽ⁿ⁺¹⁾ =f(u⁽ⁿ⁾) +cu⁽ⁿ⁾ in Ω, (3.6) u⁽ⁿ⁺¹⁾ =u⁽ⁿ⁾ on ∂Ω.

To begin to analyze this iteration we will state some useful lemmas from Pao [15], these are quoted as Lemma3.1.4, Lemma3.1.5and Lemma3.1.7.

Lemma 3.1.4 Let Lp(Ω) be the Banach space with norm ∥ u ∥Lp(Ω)=(

∫

Ω | u(ξ) |^p dξ)¹_p

andWp²(Ω)be the Sobolev space of all functions inLp(Ω)that have distributional derivativesD^lu∈ L^p(Ω)for all|l |≤2with norm∥u∥Wp²(Ω)=(

∫

Ω

∑

|l|≤2 |D^lu|^p )¹_p . Ifuis solution of (3.6)belongs toWp²(Ω), then there exists a constantK1, independent of f and h such that

||u||Wp²(Ω)≤K₁(||f ||Lp(Ω) +||h||2−¹_p).

Lemma 3.1.5 SupposeΩ∈ R^N. For any p > N the Sobolev space Wp²(Ω) is contin- uously embedded in C^1+µ( ¯Ω) withµ= 1−^N_p. That is there exists a constant K2 such that for all u∈ Wp²(Ω)

|u|^Ω1+µ^¯ ≤K₂ ||u||Wp²(Ω) .

Lemma 3.1.6 Assume f satisfies (3.5) and also let c ≥ 0. If the sequence {u⁽ⁿ⁾} defined by (3.6) is bounded in C^1+µ( ¯Ω) then {f(ξ, uⁿ)} is bounded in C^µ( ¯Ω).

A Schauder estimate is sufficient enough to show the boundedness for the solution of (3.6). In the following lemma the Schauder estimate is stated from [15].

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Lemma 3.1.7 Let f(ξ, u) ∈ C^µ( ¯Ω) and h∈ C^2+µ(∂Ω). Then the Schauder estimate for the solution u∈ C^2+µ( ¯Ω) of (3.6) is

|u|^Ω2+µ^¯ ≤K3

(|f(ξ, u)|^Ωµ^¯ +|h |^∂Ω1+µ

)

where K3 is a constant independent of u,f(ξ, u)and h. Here|.|^Ωµ^¯, |.|^∂Ω1+µ and |.|^Ω2+µ^¯

are the norms defined in spaces C^µ( ¯Ω), C^1+µ(∂Ω) and C^1+µ( ¯Ω) respectively.

The maximum principle is the basic tool to analyze monotone iterations. One useful form of this principle is the weak maximum principle which is stated below.

Lemma 3.1.8 Let w∈ C²(Ω)∩ C( ¯Ω) satisfy,

−△w+cw ≥0 in Ω, w≥0 on ∂Ω.

If c≥0 then w≥0 in Ω.

Proof: Suppose w is not positive in Ω. Since w is continuous, differentiable and non-negative on the boundary there must be a minimum negative value ofw. Assume this occurs at ξ0 in the interior of Ω. Since w has a minimum value at ξ0, we have

△w(ξ0) ≥ 0. That implies cw(ξ0) ≥ 0 which is a contradiction as c≥ 0. Hence we conclude that w≥0 in Ω.

The Arzela-Ascoli theorem is very useful for proving the convergence of se- quences. This theorem is stated as follows. The proof can be found in [11].

Theorem 3.1.9 LetX be a compact metric space and let{fn}be a uniformly bounded, equicontinuous sequence of functions on X. Then the sequence {fn} has a uniformly convergent subsequence.

Here we state another lemma from D. Gilbarg and N.S. Trudinger [10].

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Lemma 3.1.10 Let Ω be a C^q¹^+µ domain in R^N with q1 ≥1and let D be a bounded set in C^q¹^+µ( ¯Ω). Then D is precompact in C^q²^+θ( ¯Ω) if q2+θ < q1+µ.

Remark 1 Lemma3.1.10says if we take a sequence from a set D, where D is bounded set in C^q¹^+µ( ¯Ω), then this sequence has a convergent subsequence inC^q²^+θ( ¯Ω) with q2+θ < q1+µ.

The following theorem is a modified version of S.H. Lui’s [14] linearized DD approach to single domain.

Theorem 3.1.11 Let u⁽⁰⁾ = u on Ω¯ with u⁽⁰⁾ = h on ∂Ω. Consider the iteration (3.6). Then u⁽ⁿ⁾ converges to u in C²( ¯Ω), where u is a solution of equation (3.1) in A. If v is any other solution of (3.1) in A then u≤ v on Ω. If¯ u⁽⁰⁾ = ¯u on Ω¯ with

¯

u=h on ∂Ω, then the same conclusion holds except that u≥v on Ω.¯

Proof: We need to prove that the sequence ismonotonic,boundedandconverg- ing to a solution. We will prove monotonicity and boundness by induction.

From (3.6) forn = 0 we can write

− △u⁽¹⁾+cu⁽¹⁾ =f(u⁽⁰⁾) +cu⁽⁰⁾, (3.7) and u⁽¹⁾ =u⁽⁰⁾ on∂Ω.

Because u⁽⁰⁾ =u and u is a subsolution of (3.1) then

− △u⁽⁰⁾ ≤f(u⁽⁰⁾).

Adding cu⁽⁰⁾ on both sides of the above inequality, we get

− △u⁽⁰⁾+cu⁽⁰⁾ ≤f(u⁽⁰⁾) +cu⁽⁰⁾. (3.8)

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Using this inequality in (3.7) and rearranging

− △u⁽⁰⁾+cu⁽⁰⁾+△u⁽¹⁾−cu⁽¹⁾ ≤0 on Ω.

Rearranging and multiplying the above result by −1, we have

− △(u⁽¹⁾−u⁽⁰⁾) +c(u⁽¹⁾−u⁽⁰⁾)≥0 on Ω.

Since u⁽¹⁾ =u⁽⁰⁾ on∂Ω that isu⁽¹⁾−u⁽⁰⁾= 0 on ∂Ω, we conclude that u⁽¹⁾−u⁽⁰⁾ ≥0 oru⁽¹⁾ ≥u⁽⁰⁾ on Ω by Lemma3.1.8. As u⁽¹⁾ ≥u⁽⁰⁾ on Ω we can write from[A1] that

−c(u⁽¹⁾−u⁽⁰⁾)≤f(u¹)−f(u⁰), that is

f(u¹)≥f(u⁰)−c(u⁽¹⁾−u⁽⁰⁾). (3.9) Now from (3.7) we have

− △u⁽¹⁾+cu⁽¹⁾ =f(u⁽⁰⁾) +cu⁽⁰⁾. Adding −cu⁽¹⁾ on both sides, we obtain

− △u⁽¹⁾ =f(u⁽⁰⁾) +cu⁽⁰⁾−cu⁽¹⁾. From (3.9) we see

− △u⁽¹⁾ ≤f(u⁽¹⁾).

Adding −f(u⁽¹⁾) on both sides, we get

− △u⁽¹⁾−f(u⁽¹⁾)≤0, and we conclude u⁽¹⁾ is a subsolution of (3.1).

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Suppose that for some n, u⁽ⁿ⁾ ≥ u⁽ⁿ⁻¹⁾ and u⁽ⁿ⁾ is a subsolution of (3.1). We will prove the same is true for n+ 1. From (3.6) we can write for n

− △u⁽ⁿ⁺¹⁾+cu⁽ⁿ⁺¹⁾ =f(u⁽ⁿ⁾) +cu⁽ⁿ⁾. (3.10) Now u⁽ⁿ⁾ is a subsolution so

− △u⁽ⁿ⁾−f(u⁽ⁿ⁾)≤0.

Adding cu⁽ⁿ⁾+f(u⁽ⁿ⁾) on both sides, we get

− △u⁽ⁿ⁾+cu⁽ⁿ⁾≤f(u⁽ⁿ⁾) +cu⁽ⁿ⁾, and with the help of (3.10) we can write

− △u⁽ⁿ⁾+cu⁽ⁿ⁾≤ −△u⁽ⁿ⁺¹⁾+cu⁽ⁿ⁺¹⁾. Rearranging and multiplying by −1, we get

−(△u⁽ⁿ⁺¹⁾− △u⁽ⁿ⁾) +c(u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)≥0.

By the construction of iteration u⁽ⁿ⁺¹⁾ = u⁽ⁿ⁾ on ∂Ω that is u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾ = 0 on ∂Ω hence we conclude by Lemma3.1.8that u⁽ⁿ⁺¹⁾≥u⁽ⁿ⁾ on Ω.

Now we will show that u⁽ⁿ⁺¹⁾ is a subsolution. As u⁽ⁿ⁺¹⁾ ≥ u⁽ⁿ⁾ on Ω we can write from [A1] that

−c(u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾)≤f(uⁿ⁺¹)−f(uⁿ), that is

f(uⁿ⁺¹)≥f(uⁿ)−c(u⁽ⁿ⁺¹⁾−u⁽ⁿ⁾). (3.11)

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Now from (3.6) we have

− △u⁽ⁿ⁺¹⁾+cu⁽ⁿ⁺¹⁾ =f(u⁽ⁿ⁾) +cu⁽ⁿ⁾. Adding −cu⁽ⁿ⁺¹⁾ on both sides we get

− △u⁽ⁿ⁺¹⁾ =f(u⁽ⁿ⁾) +cu⁽ⁿ⁾−cu⁽ⁿ⁺¹⁾. By (3.11) we can write

− △u⁽ⁿ⁺¹⁾≤f(u⁽ⁿ⁺¹⁾).

Adding −f(u⁽ⁿ⁺¹⁾) on both sides we get

− △u⁽ⁿ⁺¹⁾−f(u⁽ⁿ⁺¹⁾)≤0.

Hence u⁽ⁿ⁺¹⁾ is a subsolution of (3.1).

We have proved that u⁽ⁿ⁾ is monotonically increasing from subsolution u and the sequence {u⁽ⁿ⁾} is bounded below by u in Ω. Now we need to show that for all n∈ N,u⁽ⁿ⁾ ≤u.¯

From (3.7) we know

− △u⁽¹⁾+cu⁽¹⁾ =f(u⁽⁰⁾) +cu⁽⁰⁾. (3.12) Also we know ¯u is a supersolution, so

− △u¯−f(¯u)≥0.

Adding f(¯u) +c¯u on both sides of the above inequality, we get

− △u¯+c¯u≥f(¯u) +c¯u. (3.13)

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Now subtracting (3.12) from (3.13) we have

− △(¯u−u⁽¹⁾) +c(¯u−u⁽¹⁾)≥f(¯u)−f(u⁽⁰⁾) +c(¯u−u⁽⁰⁾). (3.14) By property [A1] we know −c(¯u−u⁽⁰⁾)≤f(¯u)−f(u⁽⁰⁾), so (3.14) implies

− △(¯u−u⁽¹⁾) +c(¯u−u⁽¹⁾)≥0.

As ¯u≥u⁽¹⁾ =h on ∂Ω so by Lemma3.1.8we conclude that ¯u≥u⁽¹⁾ on Ω.

Assume that for some n, ¯u ≥ u⁽ⁿ⁾ we will prove it is true for n+ 1, that is

¯

u ≥ u⁽ⁿ⁺¹⁾. Now using the definition of equation for u⁽ⁿ⁺¹⁾ and subtracting it from (3.13) we get

− △(¯u−u⁽ⁿ⁺¹⁾) +c(¯u−u⁽ⁿ⁺¹⁾)≥f(¯u)−f(u⁽ⁿ⁾) +c(¯u−u⁽ⁿ⁾). (3.15) By property [A1] we know −c(¯u−u⁽ⁿ⁾)≤f(¯u)−f(u⁽ⁿ⁾), so (3.15) implies

− △(¯u−u⁽ⁿ⁺¹⁾) +c(¯u−u⁽ⁿ⁺¹⁾)≥0.

As ¯u ≥ u⁽ⁿ⁺¹⁾ = h on ∂Ω, Lemma3.1.8ensures that ¯u ≥ u⁽ⁿ⁺¹⁾ on Ω. Hence we conclude that

u≤u⁽ⁿ⁾ ≤u¯ on ¯Ω, n≥0. (3.16) Since the sequence is bounded above and monotonic, the point wise limits exist in ¯Ω, that is

n→∞lim u⁽ⁿ⁾(ξ) = u(ξ).

Now we want to show that {u⁽ⁿ⁾} converges to a solution of the PDE (3.1) in C²( ¯Ω). To start we show that{u⁽ⁿ⁾}is uniformly bounded inC^2+µ( ¯Ω). From the point

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wise convergence of {u⁽ⁿ⁾} and the continuity property of f, the sequence {f(u⁽ⁿ⁾)} converges point wise to f(u) as n → ∞. Also we know u(ξ) ≤u⁽ⁿ⁾(ξ)≤ u(ξ) for all¯ n∈ N and for all ξ∈Ω, where ¯¯ Ω is bounded domain. So, there exist a maximum of

¯

u(ξ) say M₁ and a minimum of u(ξ) say M₂ on ¯Ω such that M₂ ≤ u⁽ⁿ⁾(ξ) ≤ M₁ for all n∈ N. As f is H¨older continuous ¯Ω so f is continuous on ¯Ω. Hence there exists a constant C₁ such that

||f(u⁽ⁿ⁾(ξ))||L∞≤C1 for all n ∈ N and for all ξ ∈Ω.¯

Now

||f(u⁽ⁿ⁾(ξ))||Lp=(∫

Ω

|f(u⁽ⁿ⁾(ξ))|^p dξ)¹_p

≤(

||f(u⁽ⁿ⁾(ξ))||^p_L_∞|Ω|)¹_p ,

that is ||f(u⁽ⁿ⁾(ξ)) ||Lp≤C₂, where C₂ =(

|| f(u⁽ⁿ⁾(ξ))||^p_L_∞|Ω |)¹_p

which is finite.

Hence {f(u⁽ⁿ⁾)} is uniformly bounded in Lp(Ω) for every p ≥ 1. Hence by Lemma 3.1.4we know {u⁽ⁿ⁾}is uniformly bounded in Wp²(Ω). We choosep > N so that µ= 1− ^N_p, then by Lemma3.1.5, {u⁽ⁿ⁾} is uniformly bounded in C^1+µ( ¯Ω). Furthermore by Lemma3.1.6, {f(u⁽ⁿ⁾)}is uniformly bounded inC^µ( ¯Ω). So we conclude by Lemma 3.1.7that {u⁽ⁿ⁾} is uniformly bounded in C^2+µ( ¯Ω). Hence by Theorem3.1.9there exists a subsequence of {u⁽ⁿ⁾} which converges and Lemma3.1.10and Remark1 confirm that this subsequence converge in C²( ¯Ω) to u^∗. But since {u⁽ⁿ⁾} converges point wise to u and this sequence is monotonic, we have u = u^∗ and for the same reason the whole sequence {u⁽ⁿ⁾} converges in C²( ¯Ω) to u. Now f(u⁽ⁿ⁾) → f(u) as n→ ∞ which implies that u is a solution of (3.1).

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3.2 Linearized single domain method to solve u

^′′

= f (ξ, u, u

^′

)

Consider the Dirichlet problem

−u^′′+f(ξ, u, u^′) = 0, u(a) =u(b) = 0. (3.17) Cherpion developed an iterative approach in [6] and [5] to solve the BVP that has the form (3.17). Here we fully explain the proof of this theorem filling in all the details of the proof. In order to prove the theorem on a single domain first we have to prove some lemmas.

The subsolution and supersolution of (3.17) are defined in Definition3.2.1and Definition3.2.2respectively.

Definition 3.2.1 (Subsolution:) A functionα ∈ C²([a, b])is a subsolution of (3.17) if

(I) for all ξ∈[a, b], α^′′(ξ)≥f(ξ,α(ξ),α^′(ξ));

(II) α(a)≤0, α(b)≤0.

Definition 3.2.2 (Supersolution:) A function β¯ ∈ C²([a, b]) is supersolution of (3.17) if

(I) for all ξ∈[a, b], β¯^′′(ξ)≤f(ξ,β(ξ),¯ β¯^′(ξ));

(II) β(a)¯ ≥0, β(b)¯ ≥0.

Let us define some properties which we will be using in the theorem.

Definition 3.2.3 (Properties: )[C1] Let α and β¯ ∈ C²([a, b]) be the subsolution and supersolution of (3.17) such that α≤β¯and define the set D as

D ={(ξ, u, v)∈[a, b]× R²|α(ξ)≤u≤β(ξ)¯ }.

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[C2]One Sided Lipschitz Condition in u: Supposef :D → Ris a continuous function. We assume there exist M ≥0 such that for all (ξ, u1, v), (ξ, u2, v)∈D and for all u₁ ≤u₂,

f(ξ, u2, v)−f(ξ, u1, v)≤M(u2−u1).

[C3] Lipschitz Condition in u^′: Suppose f : D → R is a continuous function.

We assume there exist N ≥0 such that for all (ξ, u, v1), (ξ, u, v2) ∈D,

|f(ξ, u, v2)−f(ξ, u, v1)|≤N |v2−v1 |. (3.18) Under these properties for given subsolutionαand supersolution ¯βthe iteration schemes to solve (3.17) are defined for n= 0,1,2, ... as

−α^′′_n+1+√³

lK(ξ)α^′_n+1+lαn+1 =−f(ξ, αn, α^′_n) +√³

lK(ξ)α^′_n+lαn, (3.19) α0 =α,

and

−β_n+1^′′ +√³

lK(ξ)β_n+1^′ +lβ_n+1 =−f(ξ, β_n, β_n^′) +√³

lK(ξ)β_n^′ +lβ_n, (3.20) β0 =β,

wherel ≥0 is a constant (to be specifies later) and K(ξ)∈ C([a, b]) is antisymmetric function.

A maximum principle for a second order linear elliptic differential equation is provided in the following lemma.

Lemma 3.2.4 Assume w ∈ C², l ∈ R⁺ and K(ξ) is a continuous function on Ω.

Define the elliptic differential operator L in a bounded domain Ω with boundary ∂Ω

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as

Lw≡w^′′−√³

lK(ξ)w^′ −lw.

If Lw≤0 with w≥0 on ∂Ω then w≥0 on Ω.¯

Proof: Assume Lw is negative. By contradiction we assume that w is not positive in Ω. Since w is continuous, differentiable and non-negative on the boundary there must be a minimum negative value of w. Assume this occurs at ξ0 in the interior of Ω. So at this point w^′(ξ0) = 0 and w^′′(ξ0)≥0.

a b

w

ξ0 ξ

Figure 3.1: Contradictory shape of function w.

Evaluating the differential equation atξ0we have−lw(ξ0)≤0 which is a contradiction since w≤0. Hence we conclude that w≥0 in Ω.

Recalling the following basic result for initial value problems from [18]

Lemma 3.2.5 Consider the initial value problem

u^′′+p(ξ)u^′ +q(ξ)u=f(ξ), (3.21) u(ξ0) =u0, u^′(ξ0) = u^′₀.

If the functions p(ξ), q(ξ) and f(ξ) are continuous on [a, b] and ξ0 is any point of the interval[a, b], then there exists a unique solutionu of the problem (3.21) and that solution exists throughout the interval [a, b].

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Proof: The proof of this lemma can be found in [18] or [2].

This helps to prove the followings.

Lemma 3.2.6 If z1(ξ) and z2(ξ) for all ξ∈[a, b] satisfy z₁^′′(ξ)−√³

lK(ξ)z₁^′(ξ)−lz1(ξ) = 0, z1(a) = 0, z₁^′(a) = 1, (3.22) z₂^′′(ξ)−√³

lK(ξ)z₂^′(ξ)−lz₂(ξ) = 0, z₂(b) = 0, z₂^′(b) =−1, (3.23) respectively, where K(ξ) = −K(a+b−ξ), then z₂(ξ) =z₁(a+b−ξ).

Proof: We are given that z1(ξ) is a solution of equation (3.22) for all ξ ∈[a, b]. As ξ ∈[a, b] then a < ξ or−a >−ξ. Addinga+bon both sides of the inequality we get b > a+b−ξ. Also,b > ξ or −b <−ξ. Adding a+b on both sides of the inequality we geta < a+b−ξ. So (a+b−ξ)∈[a, b], for all ξ∈[a, b]. Now evaluating equation (3.22) at ¯ξ = (a+b−ξ)∈[a, b], we have

d²z₁( ¯ξ) d( ¯ξ)² −√³

lK( ¯ξ)dz₁( ¯ξ)

d( ¯ξ) −lz₁( ¯ξ) = 0.

As ¯ξ= (a+b−ξ), we can write d

d(a+b−ξ)

{dz₁(a+b−ξ) d(a+b−ξ)

}

−√³

lK(a+b−ξ)dz₁(a+b−ξ)

d(a+b−ξ) −lz₁(a+b−ξ) = 0.

Now differentiating using the chain rule, we get

− d dξ

{

− dz₁(a+b−ξ) dξ

} +√³

lK(a+b−ξ)dz₁(a+b−ξ)

dξ −lz₁(a+b−ξ) = 0.

Using K(ξ) =−K(a+b−ξ), we have d²z1(a+b−ξ)

dξ² −√³

lK(ξ)dz1(a+b−ξ)

d(ξ) −lz₁(a+b−ξ) = 0.

The initial condition is z₁( ¯ξ)|ξ=a^¯ = 0 or z₁(a +b − ξ)|a+b−ξ=a = 0 which implies z₁(a+b−ξ)|ξ=b = 0. Likewise ^dz_d¹_ξ^{( ¯}_¯^ξ)|ξ=a^¯ = 1 or ^dz_d(a+b¹^(a+b₋⁻_ξ)^ξ)|a+b−ξ=a = 1 and using the

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chain rule to evaluate the derivative at ξ we get −^dz¹^(a+b_dξ ⁻^ξ)|ξ=b = 1 which implies

dz1(a+b−ξ)

dξ |ξ=b =−1. So z1(a+b−ξ) is a solution of (3.23), hence by the uniqueness Lemma3.2.5we conclude z₂(ξ) = z₁(a+b−ξ).

In the following lemma we prove the solutions of equations (3.22) and (3.23) are always positive.

Lemma 3.2.7 Assume K(ξ)∈ C([a, b]) is such that K(a)>0 and for all ξ ∈[a, b], K(ξ) =−K(a+b−ξ). Furthermore, assume z1 is the solution of equation

z₁^′′−√³

lK(ξ)z₁^′ −lz1 = 0, z1(a) = 0, z₁^′(a) = 1. (3.24) Then for ξ ∈ (a, b], z1(ξ) > 0 and z₁^′(ξ) > 0. Furthermore we have z2 > 0 and z₂^′(ξ)<0, where z2 is a solution of equation

z₂^′′−√³

lK(ξ)z₂^′ −lz2 = 0, z2(b) = 0, z₂^′(b) = −1. (3.25) Proof: We proceed by contradiction. Assume there exists a point ξ0 ∈ (a, b] such thatz^′₁(ξ0) = 0. As the function z1 is increasing initially, to have z₁^′(ξ0) = 0 we would needz₁^′′(ξ0)≤0, see Figure3.2. But at ξ0 we havez₁^′′(ξ0)−√³

lK(ξ0)z₁^′(ξ0)−lz1(ξ0) = 0.

a b

y

ξ0 x

Figure 3.2: Sample shape of function z₁.

This impliesz₁^′′(ξ0) = lz1(ξ0)>0, which is a contradiction. Hence the result is proved.

As z2(ξ) =z1(a+b−ξ), we also have that for all ξ ∈[a, b),z2(ξ)>0 andz₂^′(ξ)<0, because ^dz_dξ²^(ξ) =−z₁^′(a+b−ξ).

Linearized Domain Decomposition Approaches for Boundary Value Problems